Synthetic division calculator
This calculator divides polynomials by binomials using synthetic division. Additionally, the calculator computes the remainder when a polynomial is divided by x−c and checks if the divisor is a factor of dividend. The calculator shows all the steps and provides a full explanation for each step.
Divide using synthetic division $$ ( 30x^{6}-32x^{5}-17x^{4}+17x^{3}+13x^{2}+8x+34 ) \div ( x-1 ) $$
solution
The synthetic division table is:
$$ \begin{array}{c|rrrrrrr}1&30&-32&-17&17&13&8&34\\& & 30& -2& -19& -2& 11& \color{black}{19} \\ \hline &\color{blue}{30}&\color{blue}{-2}&\color{blue}{-19}&\color{blue}{-2}&\color{blue}{11}&\color{blue}{19}&\color{orangered}{53} \end{array} $$The solution is:
$$ \frac{ 30x^{6}-32x^{5}-17x^{4}+17x^{3}+13x^{2}+8x+34 }{ x-1 } = \color{blue}{30x^{5}-2x^{4}-19x^{3}-2x^{2}+11x+19} ~+~ \frac{ \color{red}{ 53 } }{ x-1 } $$explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -1 = 0 $ ( $ x = \color{blue}{ 1 } $ ) at the left.
$$ \begin{array}{c|rrrrrrr}\color{blue}{1}&30&-32&-17&17&13&8&34\\& & & & & & & \\ \hline &&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrr}1&\color{orangered}{ 30 }&-32&-17&17&13&8&34\\& & & & & & & \\ \hline &\color{orangered}{30}&&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 30 } = \color{blue}{ 30 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{1}&30&-32&-17&17&13&8&34\\& & \color{blue}{30} & & & & & \\ \hline &\color{blue}{30}&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -32 } + \color{orangered}{ 30 } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrrrrr}1&30&\color{orangered}{ -32 }&-17&17&13&8&34\\& & \color{orangered}{30} & & & & & \\ \hline &30&\color{orangered}{-2}&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ -2 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{1}&30&-32&-17&17&13&8&34\\& & 30& \color{blue}{-2} & & & & \\ \hline &30&\color{blue}{-2}&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -17 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ -19 } $
$$ \begin{array}{c|rrrrrrr}1&30&-32&\color{orangered}{ -17 }&17&13&8&34\\& & 30& \color{orangered}{-2} & & & & \\ \hline &30&-2&\color{orangered}{-19}&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ \left( -19 \right) } = \color{blue}{ -19 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{1}&30&-32&-17&17&13&8&34\\& & 30& -2& \color{blue}{-19} & & & \\ \hline &30&-2&\color{blue}{-19}&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 17 } + \color{orangered}{ \left( -19 \right) } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrrrrr}1&30&-32&-17&\color{orangered}{ 17 }&13&8&34\\& & 30& -2& \color{orangered}{-19} & & & \\ \hline &30&-2&-19&\color{orangered}{-2}&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ -2 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{1}&30&-32&-17&17&13&8&34\\& & 30& -2& -19& \color{blue}{-2} & & \\ \hline &30&-2&-19&\color{blue}{-2}&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 13 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ 11 } $
$$ \begin{array}{c|rrrrrrr}1&30&-32&-17&17&\color{orangered}{ 13 }&8&34\\& & 30& -2& -19& \color{orangered}{-2} & & \\ \hline &30&-2&-19&-2&\color{orangered}{11}&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 11 } = \color{blue}{ 11 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{1}&30&-32&-17&17&13&8&34\\& & 30& -2& -19& -2& \color{blue}{11} & \\ \hline &30&-2&-19&-2&\color{blue}{11}&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ 11 } = \color{orangered}{ 19 } $
$$ \begin{array}{c|rrrrrrr}1&30&-32&-17&17&13&\color{orangered}{ 8 }&34\\& & 30& -2& -19& -2& \color{orangered}{11} & \\ \hline &30&-2&-19&-2&11&\color{orangered}{19}& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 19 } = \color{blue}{ 19 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{1}&30&-32&-17&17&13&8&34\\& & 30& -2& -19& -2& 11& \color{blue}{19} \\ \hline &30&-2&-19&-2&11&\color{blue}{19}& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ 34 } + \color{orangered}{ 19 } = \color{orangered}{ 53 } $
$$ \begin{array}{c|rrrrrrr}1&30&-32&-17&17&13&8&\color{orangered}{ 34 }\\& & 30& -2& -19& -2& 11& \color{orangered}{19} \\ \hline &\color{blue}{30}&\color{blue}{-2}&\color{blue}{-19}&\color{blue}{-2}&\color{blue}{11}&\color{blue}{19}&\color{orangered}{53} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 30x^{5}-2x^{4}-19x^{3}-2x^{2}+11x+19 } $ with a remainder of $ \color{red}{ 53 } $.
Tutorial
What is synthetic division?
Synthetic division is, by far, the easiest and fastest method to divide a polynomial by x-c, where c is a constant. This method only works when we divide by a linear factor. Let's look at two examples to learn how we can apply this method.
Worked examples
Example 1 : Divide x2+3x-2 by x-2.
Step 1: Write the coefficients of 2x2+3x+4 into the division table.
| 2 | 3 | 4 | |
Step 2: Change the sign of a number in the divisor and write it on the left side. In this case, the divisor is x - 2 so we have to change -2 to 2.
| 2 | 2 | 3 | 4 |
Step 3: Carry down the leading coefficient
| 2 | 2 | 3 | 4 |
| 2 |
Step 4: Multiply the previous value by the left term and put the result into the next column.
| 2 | 2 | 3 | 4 |
| 4 | |||
| 2 |
Step 5: Add the last column
| 2 | 2 | 3 | 4 |
| 4 | |||
| 2 | 7 |
Step 6: Multiply previous value by left term and put the result into the next column
| 2 | 2 | 3 | 4 |
| 4 | 14 | ||
| 2 | 7 |
Step 7: Add the last column
| 2 | 2 | 3 | 4 |
| 4 | 14 | ||
| 2 | 7 | 18 |
Step 8: Read the result from the synthetic table.
| 2 | 2 | 3 | 4 |
| 4 | 14 | ||
| 2 | 7 | 18 |
The quotient is 2x + 7 and the remainder is 18.
Starting polynomial x2+3x-2 can be written as:
$$ x^2 +3x - 2 = \color{blue}{2x + 7} + \dfrac{ \color{orangered}{18} }{ x - 2 } $$Example 2:
Learn how to apply synthetic division in 60 seconds.
Example 3: Divide x4+10x+1 by x + 2.
Step 1: Write down the coefficients of x4-10x+1 into the division table. (Note that this polynomial doesn't have x3 and x2 terms, so these coefficients must be zero)
| 1 | 0 | 0 | 10 | 1 | |
Step 2: Change the sign of a number in the divisor and write it on the left side. In this case, the divisor is x+3 so we have to change +3 to -3.
| -3 | 1 | 0 | 0 | 10 | 1 |
Step 3: Carry down the leading coefficient
| -3 | 1 | 0 | 0 | 10 | 1 |
| 1 |
Multiply carry-down by left term and put the result into the next column
| -3 | 1 | 0 | 0 | 10 | 1 |
| -3 | |||||
| 1 |
ADD the last column
| -3 | 1 | 0 | 0 | 10 | 1 |
| -3 | |||||
| 1 | -3 |
Multiply last value by left term and put the result into the next column
| -3 | 1 | 0 | 0 | 10 | 1 |
| -3 | 9 | ||||
| 1 | -3 |
ADD the last column
| -3 | 1 | 0 | 0 | 10 | 1 |
| -3 | 9 | ||||
| 1 | -3 | 9 |
Multiply last value by left term and put the result into the next column
| -3 | 1 | 0 | 0 | 10 | 1 |
| -3 | 9 | -27 | |||
| 1 | -3 | 9 |
ADD the last column
| -3 | 1 | 0 | 0 | 10 | 1 |
| -3 | 9 | -27 | |||
| 1 | -3 | 9 | -17 |
Multiply last value by left term and put the result into the next column.
| -3 | 1 | 0 | 0 | 10 | 1 |
| -3 | 9 | -27 | 51 | ||
| 1 | -3 | 9 | -17 |
ADD the last column
| -3 | 1 | 0 | 0 | 10 | 1 |
| -3 | 9 | -27 | 51 | ||
| 1 | -3 | 9 | -17 | 52 |
Step 7: Read the result from the synthetic table.
| -3 | 1 | 0 | 0 | 10 | 1 |
| -3 | 9 | -27 | 51 | ||
| 1 | -3 | 9 | -17 | 52 |
The quotient is x3-3x2+9x-17 and the remainder is 52.
Starting polynomial x4 + 10x + 1 can be written as:
$$ x^4 + 10x + 1 = \color{blue}{x^3 - 3x^2 + 9x - 17} + \dfrac{ \color{orangered}{52} }{ x + 3 } $$1. Synthetic division — college algebra tutorial.
2. Basic examples on how to apply synthetic division.
3. Video tutorial on how to divide third order polynomial by the monomial.
4. Synthetic division algorithm — step-by-step approach.